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Consider a couple of real random variables $(X,Y)$ and $\mu_{X,Y}$ the induced probability distribution. Denote by $\mu_X$ and $\mu_Y$ the distributions of $X$ and $Y$.

Is it true that $$\int_{\mathbb{R}^2}f(x)\ d\mu_{X,Y}(x,y)=\int_{\mathbb{R}}f(x)d\mu_{X}\quad ?$$ This is true each time you can use Fubini theorem. For example, if the random variables are discrete or indepedent or if $(X,Y)$ admits a density with respect to the Lebesgue measure.

I don't know if it is true in the general case, I don't find any counterexample.

Thanks for helping me.

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isn't by definition $P(x)=\int P(x,y)dy$, so $\int f(x)P(x,y)dxdy=\int f(x)P(x)dx$ –  Carlo Beenakker Nov 5 '12 at 13:41
    
What is your assumption about $f$? –  Davide Giraudo Nov 5 '12 at 14:23
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